Closed-form Solutions For Free Vibration Research Articles

Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.

Exact Closed-Form Solutions for Free Vibration of Double-Beam Systems Interconnected by Elastic Supports Under Axial Forces

This paper aims to present the exact closed-form solutions for the free vibration of double-beam systems composed of two parallel beams connected by an arbitrary number of discrete elastic supports. The general solutions of the mode shapes of the double-beam system are derived employing the Laplace transform method from a perspective of the entire domain of beams without enforcement of any segmentation. A unified strategy applied to various boundary conditions is proposed to determine the independent constants involved in the general solutions, as well as the frequency equation. Numerical calculations are performed to verify the present solutions by comparing the results from the previous literature and finite element simulation, and to discuss the effects of support parameters (stiffness, location, and number) on the modal characteristics of the double-beam system in detail. Outcomes show that the support location plays a pivotal role in regulating the modal characteristics of the double-beam system; for each-order mode, there are one or more potential optimal positions to maximize the effect of the elastic support. The mode veering phenomenon is detected as the support parameters change. It is highlighted that, by introducing an amplitude similarity index, the proximity degree for the mode shapes of the two beams influenced by the support parameters can be evaluated quantitatively. The present analysis is greatly helpful to the optimal design, health monitoring, and vibration control of the double-beam system.

Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports

This paper presents exact closed-form solutions for free vibration of discretely supported Euler-Bernoulli (DEB) and Timoshenko beams (DTB) in the presence of an arbitrary number of intermediate elastic constraints. The exact eigenvalue equations and mode shapes of the DEB and DTB are derived using the generalized function method for the first time, which expands the general solution of mode shapes as a combination of the standard trigonometric/hyperbolic functions with integration constants extended to generalized functions. The second-order eigenvalue equation is formulated from the perspective of the entire domain of the beam without enforcement of any continuity conditions. Several typical constraints of interest in practical engineering are investigated and other cases can be achieved by following similar methodologies elucidated in this paper. The accuracy of present solutions is validated by numerical results obtained from the finite element method (FEM) and previous literature adopting the transfer matrix method (TMM) and Green's function method (GFM). The exiting frequency equation and mode shape of a beam with elastic end constraints are proved to be the degenerate solutions of current study. Courant's maximum-minimum principle is reproduced with detailed discussions on the support stiffnesses and positions, bimodal behavior, and sensitivity to geometrical variations. As an extended application, a new method based on rigorous mathematical deduction is proposed to efficiently determine the critical stiffness leading to the maximum fundamental frequency of a beam with multiple intermediate supports. This study is of significance to the support optimization and dynamic analysis of discretely supported beam-like structures.

Closed-form solutions for free vibration of rectangular nonlocal Mindlin plates with arbitrary homogeneous boundary conditions

In this paper, an extended separation-of-variable (SOV) method is developed to achieve the closed-form solutions for free vibration of rectangular nonlocal Mindlin plates with arbitrary homogeneous boundary conditions. With the weighted residual notion, the amplitude of the potential energy functional and the kinetic energy coefficient are derived, then the Rayleigh's principle is formulated for the nonlocal Mindlin plates. In accordance with the principle, the extended SOV method is realized. In the present method, the mode functions are in a separation-of-variable form and the natural frequencies corresponding to two-direction mode functions are considered to be independent in a mathematical sense. Numerical comparison results verify the correctness and accuracy of the present method. Moreover, the effects of nonlocal parameter, boundary conditions and thickness ratio on the free vibration are discussed comprehensively.

Free vibration of nonlocal beams: Boundary value problem and a calibration method

The paradox of abnormal frequencies exists for free vibration of nonlocal cantilever beams. This work is concerned with solving this paradox within the framework of nonlocal strain gradient theory. The inconsistences of the reported boundary conditions are updated by using the weighted residual method. Then, the closed-form solutions for free vibration of cantilever beams is obtained in terms of the inconsistences of the reported boundary value problems (BVPs) and the reformulated BVPs. The softening phenomenon is captured by the updated numerical results. A method for calibrating the size-effect parameters is firstly proposed in this work. The numerical results show that the present model is capable for capturing the size-dependent mechanical properties of materials, which exhibit either the stiffness-hardening effect or the stiffness-softening effect, depending upon the relative values of the two size-effect parameters.

Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates

This work presents the highly accurate closed-form solutions for free vibration and eigenbuckling of isotropic rectangular nanoplates with arbitrary homogeneous boundary conditions based on Eringen’s nonlocal theory and classical thin plate theory.The iterative separation-of-variable (iSOV) method based on the Rayleigh quotient, which is the most accurate closed-form solution method among all separation-of-variable methods, is used to obtain the highly accurate solutions, including the exact well-known Navier and Levy types of solutions. The highly accurate closed-form solutions for free vibration of rectangular nanoplates and for eigenbuckling of different scale rectangular plates with arbitrary homogeneous boundary conditions are achieved for the first time, and all solutions are presented in excellent explicit forms.The present solutions coincide well with analytical and numerical solutions in literature, verifying the accuracy of the present method. The influences of nonlocal parameters, boundary conditions and lengths of nanoplates on frequencies and critical buckling loads are studied, and nonlocal effects are explained in physical sense. The present solutions can be taken as the benchmarks for the validation of numerical methods, a guide in parametric design of structure, and the basis of constructing new numerical methods.

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton’s principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson’s ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

Closed-form frequency solutions for simplified strain gradient beams with higher-order inertia

This paper develops an Euler−Bernoulli beam model within the context of a simplified strain gradient theory with higher-order inertia. In contrast to the classical beam models, the proposed gradient beam models can capture the size effects by introducing not only the internal length l2 related to strain gradient but also the internal length l1 related to velocity gradient. The governing equation of motion and boundary conditions are derived by using the variational principles. The closed-form solutions for free vibrations of beams with three typical boundary conditions are obtained. Numerical results show that the choices of the higher-order boundary conditions have a minor effect on the natural frequencies of beams. In addition, the inclusion of the strain gradient parameter l2 increases the effective stiffness of beams and hence it increases the natural frequencies of beams; whereas the inclusion of the velocity gradient parameter l1 acts as an equivalent compression force in the governing equation and therefore it leads to the decrease of the natural frequencies of beams. Moreover, the significant Poisson effect on the natural frequencies of beams is observed when the thickness of the beam is comparable to the strain gradient parameter l2. The closed-form solutions for natural frequencies of beams presented in this work may serve as benchmark results for other numerical methods.

Exact compact characteristic equations and new results for free vibrations of orthotropic rectangular Mindlin plates

This paper presents correct results for a paper by Liu and Xing (2011) [1] that had errors, greatly simplifies its formulations, overcomes its numerical stability problem, and provides some new results. In their early paper, Liu and Xing obtained exact closed-form solutions for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method. The equations presented therein are correct, but the numerical values of the frequencies listed have some errors because the shear rigidities of the three-ply laminates were not correctly computed. In addition, the exact characteristic equations of the early paper can also be greatly simplified and the compact equations are presented herein. The characteristic equations in the early paper may have computational stability problem when plates become very thin. This paper overcome such computational difficulty. Some new exact frequency parameters are also included in this work.

Characteristic equations and closed-form solutions for free vibrations of rectangular mindlin plates

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots, which is the particularity of Mindlin plate theory. And the closed-form solutions, which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness, are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.

Higher-order Closed-form Solutions for Free Vibration of Laminated Composite and Sandwich Shells

Closed-form formulations of two-dimensional (2D) higher-order shear deformation theories (HOSTs) for the free vibration analysis of simply supported cross-ply laminated composite and sandwich doubly curved shells are presented. The formulation includes the Sander’s theory for doubly curved shells. Two of the HOSTs account for the effects of both transverse shear strains/stresses and the transverse normal strain/stress, while the third includes only the effects of transverse shear deformation. In these developments a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique and by solving the eigenvalue equations. Numerical results are presented for the natural frequencies of laminated composite and sandwich shallow shells. The closed-form solutions presented herein for laminated composite plates and shells are compared with the available 3D elasticity and analytical solutions and it is believed that the solutions for sandwich laminates using various higher-order displacement models will serve as benchmark in future.

Higher-order Closed-form Solutions for Free Vibration of Laminated Composite and Sandwich Shells

Closed-form formulations of 2-D higher-order shear deformation theories (HOSTs) for the free vibration analysis of simply supported cross-ply laminated composite and sandwich doubly curved shells are presented. The formulation includes the Sanders theory for doubly curved shells. Two of the HOSTs account for the effects of both transverse shear strains/stresses and the transverse normal strain/stress, while the third includes only the effects of transverse shear deformation. In these developments, a realistic parabolic distribution of transverse shear strains through the shell thickness is assumed. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique and by solving the eigenvalue equations. Numerical results are presented for the natural frequencies of laminated composite and sandwich shallow shells. The closed-form solutions presented herein for laminated composite plates and shells are compared with the available 3-D elasticity and analytical solutions and it is believed that the solutions for sandwich laminates using various higher-order displacement models will serve as the benchmark in future.

Analytical modal analysis of thin-film flat lenses

The free vibrations of circular and elliptical thin-film lens are investigated. In particular, linear closed-form solutions for free vibrations of these structures were achieved and modal analysis was performed. The vibration response of the thin-film membranes were mathematically modelled using the Mathieu equation. Numerical results for various nodal diameters were computed. For the limited case, when an elliptical lens becomes circular, an excellent comparison was established with the available analytical solution. Experimental analyses were conducted to determine the effects of various parameters, such as material properties, membrane pre-strain rate, and the geometry, on natural frequency and mode shapes of these structures. The comparison verified the adequacy of linear solutions to predict the dynamic response of thin-film lenses.

Vibratory characteristics of flexural non-uniform Euler–Bernoulli beams carrying an arbitrary number of spring–mass systems

A new exact method for the analysis of free flexural vibrations of non-uniform multi-step Euler–Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and two-degree-of-freedom spring–mass systems is presented in this paper. The closed-form solutions for free vibrations of non-uniform Euler–Bernoulli beams are derived for five important cases. Then, using the massless equivalent springs to replace the spring–mass systems and the fundamental solutions developed in this paper, the frequency equation for free flexural vibrations of a multi-step non-uniform beam with any kind of support configurations and carrying an arbitrary number of spring–mass systems can be conveniently established from a second-order determinant. The proposed method is computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures.